Chapter 5 Theoretical techniques
5.2 The GAMESS approach to theoretical Compton profiles
The General Atomic and Molecular Electronic Structure System (gamess)54 is a computa- tional code which has shown itself to be useful in the study of spin momentum densities.
Koizumi et al. published extensively on the La2−2xSr1+2xMn2O7 system, combining experi-
mental magnetic Compton scattering data with gamess calculations to better understand
the electronic structure of this ferromagnetic perovskite55–58. Qureshi et al. used the same approach in their work on Co3V2O859. In both cases, the authors were able to determine
which of the TM 3d t2g and eg orbitals were populated and reconstruct the two-dimensional
spin momentum densities (2-D SMD). Recently, the Compton Scattering group at Warwick
has obtained the gamess-us code (a freely available variant) and applied its capabilities in
these systems already yielding interesting results.
GAMESS theory and operation
gamess is anab initio quantum chemistry package. For a given molecule or cluster of atoms, limited in number only by computational considerations, the gamess code can produce a
set of self-consistent molecular wavefunctions by combining atomic orbitals (AOs). Each
molecular wavefunction, Ψ(r, θ, φ), is given by
Ψ(r, θ, φ) =X
k
ckψk(r, θ, φ), (5.6)
where ψk(r, θ, φ) are the AOs of the atoms involved in the calculation and ck are their
respective weighting factors. The primary aim of a linear combination of orbitals (LCAO)
package like gamessis the determination of the setck, enabling the full molecular orbitals to
be constructed. This is achieved with iterative computation, reducing the total system energy
while considering the changes in energy and density - as in DFT methods, these changes are
reduced to some predefined threshold when self-consistency is reached.
The basis set
The AOs are specified by the choice of basis set, of which there are many types available,
suited to different types of atoms (transition metals, rare-earths, actinides, etc.). In this
work, the basis sets were obtained from the online EMSL Basis Set Exchange60. Commonly used basis sets are those of triple-zeta-valence (TZV) type, which go beyond the simplest
approximations by allowing polarisation and some expansion/contraction of the AOs. A TZV
basis set was used by Koizumi et al. in their work, mentioned above, and so was employed in
the TbMnO3 work detailed here in section7.3. This aside, all AOs possess a similar form to
that given in equation2.5, the atomic solution of the Schr¨odinger equation.
Sampling the 3-D atomic/molecular wavefunctions
The molecular wavefunctions of gamesswere interpreted by a program calledmacmolplt61, a graphics package developed for plotting 3-D molecular structures, as well as 2-D and 3-D
wavefunctions and electron densities. Using the functionality of macmolplt, the real-space
ψ( r)
(s)
χ
B
B
n
n
J
J
(q)
3-D FT 1-D FT 1-D FT Proj. Diff. Int. HT AC 3-D FT Ωp 2 l l Ωs Ωq(p)
(p)
(s)
(q)
(p)
Figure 5.1: Reversible (blue) and irreversible (red) mathematical operations for various real-space and momentum-space expressions relating to Compton scattering. FT is a Fourier transformation, HT is a Hankel transformation and AC is autocorrelation. | |2 is the square modulus, Proj. is the projection into a single dimension, and Int. and Diff. are single-dimension integration and differentiation, respectively. Theh iΩ operations represent obtaining spherical averages through double integration. Reproduced from the work of Saenz et al.37 - for more information, refer to the original text.
atom or atomic cluster of interest, where nis simply an integer. It was essential at this stage
to ensure that the wavefunctions were effectively zero at the edges of the box; if electron
density was cut off, the corresponding Compton profiles would be unrepresentative of the
simulated molecular orbitals. For this reason, an investigation was performed to find the
optimum box size, and subsequently a 99×99×99 box of size 9 ˚A was used to sample all
of the relevant wavefunctions. The most spatially delocalised electron densities examined
in this work, those of the hybridised 3d/2sp orbitals of TbMnO3, were seen to diminish to
zero within ∼2/3 of the sampling volume. This molecular orbital sampling strategy provided
real-space, and ultimately momentum-space, resolution sufficient to obtain theoretical profiles
comparable with experimental data.
Obtaining the Compton profiles
Manipulation of the molecular wavefunctions was performed using matlab. Once a real-space
wavefunction was obtained, its Fourier transform was taken in order to obtain χ(p), its
momentum-space equivalent. The modulus of this was squared, giving n(p), the 3-D electron
momentum density (3-D EMD). Finally, n(p) was projected into a single dimension by
Compton profiles p x(a.u.) py (a.u .) −10 −5 0 5 10 −10 −5 0 5 10 Co 3dxywavefunction px(a.u.) pz (a.u .) −10 −5 0 5 10 −10 −5 0 5 10
2-D EMD in the planepx-pz
2-D EMD in the planepx-py
Fourier transform, square modulus and projection onto a plane.
Integration along the desired direction.
Figure 5.2: Obtaining electron densities and Compton profiles from the result of a gamess calculation. The example here is the 3dxy orbital of a Co atom surrounded by an arrangement
for each molecular wavefunction was a two-column dataset of momentum and projected
electron density, the Compton profile. These mathematical processes are defined concisely in
equations3.8a,3.8b,3.9a and3.9b, as well as being illustrated in figure5.1. The Compton
profiles of the molecular orbitals possessed arbitrary momentum scales, and so were convoluted
with a resolution function of 0.4 a.u. (see section 4.2) and interpolated onto experimental
momentum scales to allow realistic comparison, and fitting, to data. Figure 5.2 illustrates the
process of obtaining Compton profiles from the result of agamess calculation - the example
given is the 3dxy orbital of a Co atom surrounded by La atoms, as in the cubic perovskite
system LaCoO3, a system previously studied by the Compton Scattering group.